Constrained systems and statistical distribution

Theoretical advances in the statistical description of non-Hamiltonian systems [Tuckerman, Mundy, and Martyna, Europhys. Lett. 45, 149 (1999)] have recently led us to rethink the definition of the phase-space probability by using the proper invariant measure. Starting from this point of view, we will derive the statistical distribution of constrained systems, considered as non-Hamiltonian, in Cartesian coordinates, in a way independent from the standard Lagrangian treatment. Furthermore, we will analyze the statistical distribution of the Nose-Hoover isothermal (canonical or NVT) dynamics, considering with care the conservation laws hidden in the evolution equations. Consequently, we will correct the equations of motion in order to obtain the proper statistical ensemble. The isobaric-isothermal (constant pressure or NpT) form of the Nose-Hoover dynamics is then considered as a nontrivial extension of the described procedure and, similarly to the NVT case, we will correct the equations of motion with respect to previous versions. The case of a constrained system coupled to a thermostat and a piston is easily handled by means of the same formalism, whereas the Lagrangian treatment becomes involved.